An introduction to the basics of dephasing Florian Marquardt ([email protected], University of Basel, Switzerland) 10th August 2001 Abstract I offer a simple introduction to the basics of dephasing (or “decoherence”), i.e. the process by which quantum- mechanical interference is destroyed. Emphasis is placed on the physical principles and concrete examples, rather than the formalism. Spins in a spatially varying field Oscillations fading away An example for this kind of behaviour is an ensemble of spins in an inhomogeneous magnetic field (pictured be- Oscillations are ubiquitous in physics, and so is their de- low), when the field strength differs between the places cay, shown in the picture below: of the spins. Since the field controls the precession fre- quency, the average magnetization shows decaying oscil- lations, just like it has been explained above. This effect is important in nuclear and electronic spin resonance exper- iments and is one of the first examples where “dephas- ing” was analyzed for a quantum-mechanical two-level system, the spin. To observe it, one first applies a Rabi- pulse of an (additional) oscillating magnetic field in such a way that all the spins are flipped from their initial di- rection (pointing along the external field) into the plane Maybe this curve just represents the amplitude of a pen- perpendicular to it. Then the components of the magne- dulum, which loses its energy due to friction and finally tization within that plane will show the decaying pattern comes to a rest. Then it would have little to do with the displayed above. phenomenon which we call “dephasing” and which this short introduction is about. But maybe the graph really shows a superposition of many oscillatory curves, each B B one of them having a slightly different frequency - see the next picture (where the frequencies are distinguished by appropriate colors). Initially all the oscillations start with M the same phase, but, with time progressing, they “get out of phase”, so their average value decays. M 1 The spin is a particularly nice example to illustrate the ef- Fluctuations which merely change the phase fects of dephasing. This is because a spin lying in the xy plane is described by a quantum-mechanical super- position of the states “spin up” and “spin down” along Now, you should not start to believe that a spin will not be the z-direction, both having equal weights (probabilities). dephased by the thermal fluctuations of the magnetic field. The phase-factor ¢¡¤£ which appears in this superposition This impression may arise here merely because, up to now, we have assumed that only the “z-component” of the takes on a direct geometrical meaning, since the phase ¥ magnetic field fluctuates, which is the one that controls is equal to the angle ¥ of the spin in the xy plane. The the precession frequency, via the energy difference be- ¥ ¦ time-rate of change in ¥ , i.e. the precession frequency , is given by the energy difference between the up and down tween “up” and “down”. Such a sort of external influence, states, which is proportional to the applied magnetic field. which only changes the energies of different quantum- Of course, this kind of picture can be applied to any two- mechanical states, is called “diagonal coupling”, because level system, even if the two states have a rather abstract the quantum-mechanical operator representing the action meaning. of the external field is diagonal in the eigenbasis of the system to be dephased, i.e. the spin. It is the simplest kind of coupling (at least for calculations). Fluctuations in time Besides fluctuations in space, another source of dephasing in an ensemble of spins may be fluctuations of the mag- Fluctuations that can flip the spin netic field in time. In such a case, the phase of the pre- cessing spin advances with a rate which is constant only on the average. Sometimes the phase will grow faster than However, the fluctuating magnetic field usually will have usual, sometimes it will lag behind, giving rise to a fluc- components that point into directions other than the “spe- tuating time-evolution of the magnetization itself. Again, cial” z-axis which is selected by the strong, externally ap- a decaying oscillatory pattern will result if one averages plied static field. These other components can even tip this over many spins, each of which is subject to different the spin out of the xy-plane. And, furthermore, for them time-dependent fluctuations. Alternatively, one could run the argument about the cancellation of “positive and nega- an experiment many times on a single system and average tive” fluctuations does not work anymore, so that, indeed, over the outcomes of many runs. the thermally fluctuating magnetic field will randomize the phase of the spin and its angle with respect to the z- At this point, it is interesting to note that not all kinds axis. This coupling is called “nondiagonal” (as you may of fluctuations will lead to a complete decay of the os- have guessed). It can flip an “up” spin into a “down” spin cillations. Maybe positive fluctuations of the magnetic and vice versa, thereby changing the occupation proba- field (increasing its value above average) are followed by bilities of “up” and “down”. Since these states have dif- fluctuations pointing in the other direction, which tend to ferent energies (when they are in the static external field), cancel the effect on the precessing spin. Then, although an energy exchange is involved that takes place between the phase shows some jitter around its average value, the the spin (the “system”) and the fluctuating field, which is deviations from that value cannot increase with time. A often called a “bath” in the general context of dephasing. complete decay can result only if positive and negative The name “bath” derives, of course, from the heat bath fluctuations do not cancel. In that case, the phase may per- considered in thermodynamics, which is held at a given form a kind of random walk, deviating ever more strongly temperature and drives a small system into equilibrium from its average value. However, it is not uncommon for when coupled to it. We will come back later to discuss the fluctuations to be of the first, “harmless” variety: For the distinction between dephasing and the more common example, the thermal fluctuations of the magnetic field in effects of heating and energy dissipation that are also pro- free space will be exactly of this type! duced by a heat bath. 2 Fluctuations are harmless when too fast or too slow frequency spectrum of the bath matters so much in de- phasing. Of course, the strength of the fluctuations is important in determining how fast the spin is dephased. But even more important than the strength is the frequency spectrum. If Spontaneous emission of energy into the “bath”: the the fluctuations are very fast, they tend to average out and example of the atom their effect is relatively small: The dephasing time will be long. In nuclear magnetic resonance, this applies es- Up to now, the fluctuating field has been treated as if it pecially to molecules in a liquid. Since their surrounding were imposed on the spin as a kind of classical (but ran- changes when they move around, the magnetic field act- dom) external force. What has been neglected is the back- ing on a nuclear spin inside the molecule fluctuates rather reaction of the system (spin) onto the bath. This is im- rapidly. That, in turn, increases the dephasing time and portant, since it is the basis of energy dissipation - and therefore leads to a sharpening of the transition line in it also leads to dephasing, as we will see. In discussing the NMR spectrum, compared to the situation in a solid. this, let us take an atom as an example, instead of the This effect is called “motional narrowing”. One can un- spin. Although even a fluctuating classical electromag- derstand it better if one realizes that the phase performs netic field is able to de-excite the atom (“induced emis- a kind of random walk due to the fluctuating field. The sion”), it is equally likely to pump energy into it. There- total average distance travelled by a random walker gets fore, in the long run the atom would have an equal popula- smaller when the steps themselves get smaller both in du- tion of ground state and excited states, which corresponds ration and in size. This corresponds to the rapid fluctua- to equilibrium at infinite temperature. Only because there tions of the magnetic field acting on a spin in a liquid. is the extra energy dissipation provided by spontaneous emission of photons, will there be a difference in the pop- In the case of NMR, the magnetic field produced by the ulations of the atom’s energy levels. This is especially spin of a neighboring proton corresponds to a frequency important at low temperatures - and for an atom, room shift of about a few tens of kilohertz. Therefore, if there temperature is practically zero, when we regard the tran- were no fluctuations in the positions and orientation of sitions between the ground state and excited levels which neighboring spins, the dephasing time would be less than have a few electron volts of energy! Then, energy can a millisecond. Usually, however, it is rather on the order only be emitted by the atom into the field. This falls into of tens of milliseconds up to a second, owing to this aver- the category of a “nondiagonal” coupling between bath aging phenomenon. and system - i.e.